MATH 17100 28446
Test 4 solutions
12/02/ 2009
1. Consider the system of equations
2 x1 x 2 4 x 3 4 x 4 1 x1 2 x 2 x 3 4 x 4 2
,
a) Rewriting the system in matrix form Ax b , defining the new quantities involved.
x1 x2 4 1 , x , b x3 4 2 x4
2 A 1
1 2
4 1
MATH 17100 28446
Test 1solution
09/21/ 2009
H (t ) 4t
2
1. (a) Find the domain and sketch the graph of the function H given by
H (t ) 4t
2
2t
.
2t
(2 t )(2 t ) 2t
2 t, t 2
D omain of H is ( , 2) (2, )
Graph of H is the same as the graph of the equation
MATH 17100 28446
Test 2 solution
10/07/ 2009
1. Given the points
A (1, 2, 3) , B (1, 3, 6), C (3, 8, 6) and D (3, 7, 3) ,
(a) Find the distance between A and B Distance between A and B is
A B 1 1, 3 2, 6 3 0,1, 3 0 1 3
2 2 2
10
(b) Find a vector equation
LINEAR ALGEBRA
Paul Dawkins
Linear Algebra
Table of Contents
Preface . ii Outline . iii Systems of Equations and Matrices . 1
Introduction . 1 Systems of Equations . 3 Solving Systems of Equations. 15 Matrices .
MATH 17100 28446
11/16/2009
Determinants
The determinant of a square matrix A is a scalar denoted by det A or A obtained as follows: The determinant of the 11 matrix A = [a11 ] is= det[a11 ] a11 . det A =
a The determinant of the 2 2 matrix A = 11 a21
a12
MATH 17100 28446
11/18/2009
Inverse of a Matrix
Given a matrix A, suppose there exists matrix B such that
AB BA I . =
Evidently, this is possible, from conformability and compatibility conditions, only if both A and B are square and of the same order, n n
MATH 17100 28446
11/23/2009
Eigenvalues and eigenvectors of a Matrix
Given a square matrix A, and a vector x, the product Ax is also a vector, which, in general, has a magnitude and direction different from x. In other words, the square matrix A transform
Equation of plane is ax by cz ax 0 by 0 cz 0
y z 2 3 1 since ( a , b , c ) (0,1,1) as evidenced by the normal vector n.
For parametric representation of the plane, we take
x u , y v, z 1 v ,
The vector equation describing the plane would be
r (u , v ) u
MATH 17100 28446
11/11/2009
Varieties of Systems of Linear Equations
If in a row-reduced form of an augmented matrix, we encounter a row where all the elements except the last one is 0, i.e.
0 0 a , a 0, 0
0 0
0 0
in terms of the equation that row repre
MATH 17100 28446
11/09/2009
The row-reduced echelon form
An m n matrix is in row-reduced form if it has the following properties: (1) The first non-zero element encountered along any row is 1. Such an element is called a pivot element. (2) Along the colum
MATH 17100 28446
Quiz 14 solution
11/09/ 2009
1. Find the coefficient matrix and the augmented matrix of the following system of
equations:
2x 2 y z 1
(a) x y 2 z 3
x 3y 4z 6
2 A 1 1 2 1 3 1 2, 4
Coefficient matrix,
2 Augmented matrix, A | b 1 1
2 1 3
1
MATH 17100 28446
Quiz 16 solution
11/16/ 2009
1. Find the rank of the coefficient matrix as well as that of the augmented matrix associated
with the given systems of equations. Then solve the system of equations and express the solution as a vector: 3x +
MATH 17100 28446
Quiz 17 solution
11/18/ 2009
1. Evaluate the following determinant by using the Laplace Expansion Method
( 1st row expansion). Which row or column expansion would involve the least amount of work?
2 1 3 16 16 11 1 1 6 = 2( 1)1+1 + (1)(1)1
MATH 17100 28446
Quiz 18 solution
11/23/ 2009
1. For the matrices A and B given below, find the inverse, if it exists, by
(i) (ii) Row reduction method Adjoint method
1 1 3 3 1 1 4 3 2 det A det 1 0 3 1 1 1 0 3 23 2 2 ( 1) 3 0 1 1
A=
2 (3 3) 0
A has no i
MATH 17100 28446
11/02/2009
Matrices
Recall the vectors we defined as n-tuples, namely , ordered collections of numbers, for which we devised and defined operations of addition, multiplication by a scalar, dot-multiplication and cross-multiplication, usin
MATH 17100 28446
11/04/2009
Systems of Linear Equations and Elementary Row operations
Given the system of m linear equations in n unknowns x1 , x2 , xn : a11 x1 + a12 x2 + + a1n xn = . (1) b1 a21 x1 + a22 x2 + + a2 n xn = . (2) b2 am1 x1 + am 2 x2 + + amn